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How to use it?
- Derivative of:
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Derivative of x^3/(2*x+4)
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Derivative of (x^3-2)*(x^2+1)
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Derivative of x^2/(x-4)
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Derivative of (x^2+24)/(x+1)
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Identical expressions
- (arccos(5x))^(ln(x))
- (arc co sinus of e of (5x)) to the power of (ln(x))
- (arccos(5x))(ln(x))
- arccos5xlnx
- arccos5x^lnx
Derivative of (arccos(5x))^(ln(x))
The solution
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[src]
log(x) acos (5*x)
$$\operatorname{acos}^{\log{\left(x \right)}}{\left(5 x \right)}$$
d / log(x) \ --\acos (5*x)/ dx
$$\frac{d}{d x} \operatorname{acos}^{\log{\left(x \right)}}{\left(5 x \right)}$$
Detail solution
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Don't know the steps in finding this derivative.
But the derivative is
The answer is:
The first derivative
[src]
log(x) /log(acos(5*x)) 5*log(x) \
acos (5*x)*|-------------- - ------------------------|
| x ___________ |
| / 2 |
\ \/ 1 - 25*x *acos(5*x)/
$$\left(- \frac{5 \log{\left(x \right)}}{\sqrt{1 - 25 x^{2}} \operatorname{acos}{\left(5 x \right)}} + \frac{\log{\left(\operatorname{acos}{\left(5 x \right)} \right)}}{x}\right) \operatorname{acos}^{\log{\left(x \right)}}{\left(5 x \right)}$$
The second derivative
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/ 2 \
log(x) |/ log(acos(5*x)) 5*log(x) \ log(acos(5*x)) 10 25*log(x) 125*x*log(x) |
acos (5*x)*||- -------------- + ------------------------| - -------------- - -------------------------- + ----------------------- - ------------------------|
|| x ___________ | 2 ___________ / 2\ 2 3/2 |
|| / 2 | x / 2 \-1 + 25*x /*acos (5*x) / 2\ |
\\ \/ 1 - 25*x *acos(5*x)/ x*\/ 1 - 25*x *acos(5*x) \1 - 25*x / *acos(5*x)/
$$\left(- \frac{125 x \log{\left(x \right)}}{\left(1 - 25 x^{2}\right)^{\frac{3}{2}} \operatorname{acos}{\left(5 x \right)}} + \left(\frac{5 \log{\left(x \right)}}{\sqrt{1 - 25 x^{2}} \operatorname{acos}{\left(5 x \right)}} - \frac{\log{\left(\operatorname{acos}{\left(5 x \right)} \right)}}{x}\right)^{2} + \frac{25 \log{\left(x \right)}}{\left(25 x^{2} - 1\right) \operatorname{acos}^{2}{\left(5 x \right)}} - \frac{10}{x \sqrt{1 - 25 x^{2}} \operatorname{acos}{\left(5 x \right)}} - \frac{\log{\left(\operatorname{acos}{\left(5 x \right)} \right)}}{x^{2}}\right) \operatorname{acos}^{\log{\left(x \right)}}{\left(5 x \right)}$$
The third derivative
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/ 3 2 \
log(x) | / log(acos(5*x)) 5*log(x) \ 375 2*log(acos(5*x)) / log(acos(5*x)) 5*log(x) \ /log(acos(5*x)) 25*log(x) 10 125*x*log(x) \ 250*log(x) 125*log(x) 15 75 9375*x *log(x) 1875*x*log(x) |
acos (5*x)*|- |- -------------- + ------------------------| - ------------------------ + ---------------- + 3*|- -------------- + ------------------------|*|-------------- - ----------------------- + -------------------------- + ------------------------| - ------------------------- - ------------------------ + --------------------------- + ------------------------- - ------------------------ - ------------------------|
| | x ___________ | 3/2 3 | x ___________ | | 2 / 2\ 2 ___________ 3/2 | 3/2 3/2 ___________ / 2\ 2 5/2 2 |
| | / 2 | / 2\ x | / 2 | | x \-1 + 25*x /*acos (5*x) / 2 / 2\ | / 2\ 3 / 2\ 2 / 2 x*\-1 + 25*x /*acos (5*x) / 2\ / 2\ 2 |
\ \ \/ 1 - 25*x *acos(5*x)/ \1 - 25*x / *acos(5*x) \ \/ 1 - 25*x *acos(5*x)/ \ x*\/ 1 - 25*x *acos(5*x) \1 - 25*x / *acos(5*x)/ \1 - 25*x / *acos (5*x) \1 - 25*x / *acos(5*x) x *\/ 1 - 25*x *acos(5*x) \1 - 25*x / *acos(5*x) \-1 + 25*x / *acos (5*x)/
$$\left(- \frac{9375 x^{2} \log{\left(x \right)}}{\left(1 - 25 x^{2}\right)^{\frac{5}{2}} \operatorname{acos}{\left(5 x \right)}} - \frac{1875 x \log{\left(x \right)}}{\left(25 x^{2} - 1\right)^{2} \operatorname{acos}^{2}{\left(5 x \right)}} - \left(\frac{5 \log{\left(x \right)}}{\sqrt{1 - 25 x^{2}} \operatorname{acos}{\left(5 x \right)}} - \frac{\log{\left(\operatorname{acos}{\left(5 x \right)} \right)}}{x}\right)^{3} + 3 \cdot \left(\frac{5 \log{\left(x \right)}}{\sqrt{1 - 25 x^{2}} \operatorname{acos}{\left(5 x \right)}} - \frac{\log{\left(\operatorname{acos}{\left(5 x \right)} \right)}}{x}\right) \left(\frac{125 x \log{\left(x \right)}}{\left(1 - 25 x^{2}\right)^{\frac{3}{2}} \operatorname{acos}{\left(5 x \right)}} - \frac{25 \log{\left(x \right)}}{\left(25 x^{2} - 1\right) \operatorname{acos}^{2}{\left(5 x \right)}} + \frac{10}{x \sqrt{1 - 25 x^{2}} \operatorname{acos}{\left(5 x \right)}} + \frac{\log{\left(\operatorname{acos}{\left(5 x \right)} \right)}}{x^{2}}\right) - \frac{125 \log{\left(x \right)}}{\left(1 - 25 x^{2}\right)^{\frac{3}{2}} \operatorname{acos}{\left(5 x \right)}} - \frac{250 \log{\left(x \right)}}{\left(1 - 25 x^{2}\right)^{\frac{3}{2}} \operatorname{acos}^{3}{\left(5 x \right)}} - \frac{375}{\left(1 - 25 x^{2}\right)^{\frac{3}{2}} \operatorname{acos}{\left(5 x \right)}} + \frac{75}{x \left(25 x^{2} - 1\right) \operatorname{acos}^{2}{\left(5 x \right)}} + \frac{15}{x^{2} \sqrt{1 - 25 x^{2}} \operatorname{acos}{\left(5 x \right)}} + \frac{2 \log{\left(\operatorname{acos}{\left(5 x \right)} \right)}}{x^{3}}\right) \operatorname{acos}^{\log{\left(x \right)}}{\left(5 x \right)}$$
The graph